Geometry (A): READ: Proportionality Relationships Reading

Proportionality Relationships

Learning Objectives

Identify proportional segments when two sides of a triangle are cut by a segment parallel to the third side.
Divide a segment into any given number of congruent parts.

Introduction

We’ll wind up our study of similar triangles in this section. We will also extend some basic facts about similar triangles to dividing segments.

Dividing Sides of Triangles Proportionally

Think about a midsegment of a triangle. A midsegment is parallel to one side of a triangle, and that it divides the other two sides into congruent halves (because the midsegment connects the midpoints of those two sides). So the midsegment divides those two sides proportionally.

Example 1

Explain the meaning of "the midsegment divides the sides of a triangle proportionally."

Suppose each half of one side of a triangle is $x\;\mathrm{units}$ long, and each half of the other side is $y\;\mathrm{units}$ long.

One side is divided in the ratio $x: x$ , the other side in the ratio $y: y$ Both of these ratios are equivalent to $1:1$ and to each other.

We see that a midsegment divides two sides of a triangle proportionally. But what about some other segment?

Tech Note - Geometry Software

Use your geometry software to explore triangles where a line parallel to one side intersects the other two sides. Try this:

1. Set up $\triangle ABC$ .

2. Draw a line that is parallel to $\overline{AC}$ and that intersects both of the other sides of $\triangle ABC$ .

3. Label the intersection point on $\overline{AB}$ as $D$ ; label the intersection point on $\overline{CB}$ as $E$ .

Your triangle will look something like this.

$DE$ parallel to $AC$

4. Measure lengths and calculate the following ratios.

$\frac{AD} {DB} =$ ______ and $\frac{CE} {EB} =$ ______

5. Compare your results with those of other students.

Different students can start with different triangles. They can draw different lines parallel to $\overline{AC}$ . But in each case the two ratios, $\frac{AD} {DB}$ and $\frac{CE} {EB}$ , are approximately the same. This is another way to say that the two sides of the triangle are divided proportionally. We can prove this result as a theorem.

Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides into proportional segments.

Note: The converse of this theorem is also true. If a line divides two sides of a triangle into proportional segments, then the line is parallel to the third side of the triangle.

Example 2

In the diagram below, $UV : NP = 3:5$ .

What is an expression in terms of $x$ for the length of $\overline{MN}$ ?

According to the Triangle Proportionality Theorem,

$\frac{3} {5} & = \frac{MU} {MU+3x} \\ 3MU+9x & = 5MU \\ 2MU & = 9x \\ MU & = \frac{9x} {2} = 4.5x \\ MN & = MU + UN = 4.5x + 3x \\ MN & = 7.5x$

There are some very interesting corollaries to the Triangle Proportionality Theorem. One could be called the Lined Notebook Paper Corollary!

Parallel Lines and Transversals

Example 3

Look at the diagram below. We can make a corollary to the previous theorem.

$k,\;m,\;n,\;p,\;r$ are labels for lines

$a,\;b,\;c,\;d$ are lengths of segments

$k,\;m,\;n$ are parallel but not equally spaced

We’re given that lines $k,m$ , and $n$ are parallel. We can see that the parallel lines cut lines $p$ and $r$ (transversals). A corollary to the Triangle Proportionality Theorem states that the segment lengths on one transversal are proportional to the segment lengths on the other transversal.

Conclusion: $\frac{a} {b} = \frac{c} {d}$ and $\frac{a} {c} = \frac{b} {d}$

Example 4

The corollary in example 3 can be broadened to any number of parallel lines that cut any number of transversals. When this happens, all corresponding segments of the transversals are proportional!

The diagram below shows several parallel lines, $k_1 , k_2 , k_3$ and $k_4$ , that cut several transversals $t_1 , t_2$ , and $t_3$ .

$k$ lines are all parallel.

Now we have lots of proportional segments.

For example:

$\frac{a} {b} = \frac{d} {e} , \frac{a} {c} = \frac{g} {i} , \frac{b} {h} = \frac{a} {g} , \frac{c} {f} = \frac{b} {e}$ , and many more.

This corollary extends to more parallel lines cutting more transversals.

Lined Notebook Paper Corollary

Think about a sheet of lined notebook paper. A sheet has numerous equally spaced horizontal parallel segments; these are the lines a person can write on. And there is a vertical segment running down the left side of the sheet. This is the segment setting the margin, so you don’t write all the way to the edge of the paper.

Now suppose we draw a slanted segment on the sheet of lined paper.

Because the vertical margin segment is divided into congruent parts, then the slanted segment is also divided into congruent segments. This is the Lined Notebook Paper Corollary.

What we’ve done here is to divide the slanted segment into five congruent parts. By placing the slanted segment differently we could divide it into any given number of congruent parts.

History Note

In ancient times, mathematicians were interested in bisecting and trisecting angles and segments. Bisection was no problem. They were able to use basic geometry to bisect angles and segments.

But what about trisection dividing an angle or segment into exactly three congruent parts? This was a real challenge! In fact, ancient Greek geometers proved that an angle cannot be trisected using only compass and straightedge.

With the Lined Notebook Paper Corollary, though, we have an easy way to trisect a given segment.

Example 5

Trisect the segment below.

Draw equally spaced horizontal lines like lined notebook paper. Then place the segment onto the horizontal lines so that its endpoints are on two horizontal lines that are three spaces apart.

slanted segment is same length as segment above picture
endpoints are on the horizontal segments shown
slanted segment is divided into three congruent parts

The horizontal lines now trisect the segment. We could use the same method to divide a segment into any required number of congruent smaller segments.

Lesson Summary

In this lesson we began with the basic facts about similar triangles the definition and the SSS and SAS properties. Then we built on those to create numerous proportional relationships. First we examined proportional sides in triangles, then we extended that concept to dividing segments into proportional parts. We finalized those ideas with a notebook paper property that gave us a way to divide a segment into any given number of equal parts.

Points to Consider

Earlier in this book you studied congruence transformations. These are transformations in which the image is congruent to the original figure. You found that translations (slides), rotations (turns), and reflections (flips) are all congruence transformations. In the next lesson we’ll study similarity transformations transformations in which the image is similar to the original figure. We’ll focus on dilations. These are figures that we zoom in on, or zoom out on. The idea is very similar to blowing up or shrinking a photo before printing it.

The following questions are for your own review. The questions are listed below for you to check your work and understanding.

Review Questions

Use the diagram below for exercises 1-5.

Given that $\overline{DB}: \overline{EC}$

Name similar triangles.

Complete the proportion.

$\frac{AB} {BC} = \frac{?} {DE}$
$\frac{AB} {AD} = \frac{?} {DE}$
$\frac{AB} {AC} = \frac{AD} {?}$
$\frac{AC} {AE} = \frac{BC} {?}$

Lines $k$ , $m$ , and $n$ are parallel.

What is the value of $x$ ?

Lines $k$ , $m$ , and $n$ are parallel, and $AB = 30$ .

What is the value of $x$ ?
What is the value of $y$ ?
Explain how to divide a segment into seven congruent segments using the Lined Notebook Paper Corollary.

Review Answers

$\triangle ABD: \triangle ACE$ or equivalent
$AD$
$BC$
$AE$
$DE$
$22.5$
$11.25$
$18.75$
Place the original segment so that one endpoint is on the top horizontal line. Slant the segment so that the other endpoint is on the seventh horizontal line below the top line. These eight horizontal lines divide the original segment into seven congruent smaller segments.

Last modified: Monday, June 28, 2010, 6:18 PM